Reference Frame Transformations (GPS)

This section presents methods for transforming points and vectors between rectangular coordinate systems. The axes of each coordinate system are assumed to be right-handed and orthogonal. Three dimensions are used throughout the discussion; however, the discussion is equally valid for

The Direction Cosine Matrix

Letrepresent a right-handed orthogonal coordinate system. Letbe a vector from the originof theframe to the point P. The representation of the vectorwith respect to frameis

axes and where

are unit vectors along the axes and

The vectorcontains the coordinates of the point P with respect to the axes ofand is the representation of the vector with respect toThe physical interpretations of the coordinates is that they are the projections of the vectoronto theaxes. For the two-dimensional x — y plane, the discussion of this paragraph is depicted in Figure 2.9.

A vector v can be defined without reference to a specific reference frame. When convenient, as discussed above, the representation ofwith respect to the axes of frameis

Eqn. (2.14) is used in derivations later in this subsection.

Figure 2.9: Two dimensional representation of the determination of the coordinates of a point P relative to the originof reference frame.

With two distinct reference framesthe same point can be represented by a different sets of coordinates in each reference frame.The remainder of this section discusses the important question of how to use the coordinates of a point in one frame-of-reference to compute the coordinates of the same point with respect to a different frame-of-reference. The transformation of point coordinates from one frame-of-reference to another will require two operations: translation and rotation.

From the above discussion,is the vector fromto P and

is the vector fromDefineas the vector fromTherefore, we have that

This equation must hold whether the vectors are represented in the coordinates of theframe or theframe.

Denote the components of vectorrelative to theframe asthe components ofrelative to theframe asand the components ofrelative to theframe asAssume thatand the relative orientation of the two reference frames are known. Then, the position of P with respect to theframe can be computed as

Because it is the only unknown term in the right hand side, the present question of interest is how to calculatebased on the available information.

Figure 2.10: Definition of the coordinates of a point P with respect to two frames-of-reference

Letrepresent the unit vectors along the axes. As discussed relative to eqn. (2.14), vectorsdefined as

represent the unit vectors in the direction of thecoordinate axes that are resolved in thereference frame. Sinceare orthonormal, so areTherefore, the matrix is an orthonormal matrix

Each element ofis the cosine of the angle between one of and one ofTo see this, consider the element in the third row second column:

whereis the angle betweenand we have used the fact that

Figure 2.11: Definition ofin eqn. (2.16).

Because each element ofis the cosine of the angle between one of the coordinate axes ofand one of the coordinate axes ofthe matrix is referred to as a direction cosine matrix:

Figure 2.11 depicts the angles .forthat define the first column ofTheangles are defined similarly. When the relative orientation of two reference frames is known, the direction cosine matrix is unique and known.

Although the direction cosine matrix has nine elements, due to the three orthogonality constraints and the three normality constraints, there are only three degrees of freedom.

Therefore,

Point Transformation

When eqn. (2.17) is substituted into eqn. (2.15) it yields the desired equation for the transformation of the coordinates of P with respect to frame 2, as represented byto the coordinates of P with respect to frame 1, as represented by

The reverse transformation is easily shown from eqn. (2.18) to be

where we have used the fact thatwhere the last equality is true due to the orthonormality of RJ. Note that the point transformation between reference systems involves two operations: translation to account for separation of the origins, and rotation to account for non-alignment of the axis.

Vector Transformation

Consider two pointsLet the vector v denote the directed line segment fromRelative tov can be described as

Eqn. (2.20) is the vector transformation between coordinate systems. This relation is valid for any vector quantity. As discussed in detail in [95], it is important to realized that vectors, vector operations, and relations between vectors are invariant relative to any two particular coordinate representations as long as the coordinate systems are related through eqn. (2.20). This is important, as it corresponds to the intuitive notion that the physical properties of a system are invariant no matter how we orient the coordinate system in which our analysis is performed.

In the discussion of this section, the two frames have been considered to have no relative motion. Issues related to relative motion will be critically important in navigation systems and are discussed in subsequent sections.

Throughout the text, the notationwill denote the rotation matrix transforming vectors from frame a to frame b. Therefore,

Matrix Transformation

In some instances, a matrix will be defined with respect to a specific frame of reference. Eqns. (2.21-2.22) can be used to derive the transformation of such matrices between frames-of-reference.

Letbe a matrix defined with respect to frame a andbe two vectors defined in frame a. Let b be a second frame of reference. Ifare related by then eqns. (2.21-2.22) show that